How do I get the "Sniper Duel" achievement in Minecraft?

Since you have a skeleton spawner, that should make it easier. What I would do is build a wall/fence around it so that skeletons are contained, and leave some holes for you to shoot through. Next, if looking through the holes does not activate the monster spawner, I would have a doorway that leads inside the spawner room to activate it, then run out. The building will ensure that the skeletons do not wander off.

To kill the skeleton, first hit it with a sword through the hole until it's one hit away from death. Then, back off 50 blocks and snipe it!

I think it also counts if you're 50 blocks above the skeleton. Build a snipers tower, wait until dark and then snipe away!